math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 9.4s

Alternatives: 16

Speedup: 1.5×

• 49.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \cosh im \cdot \sin re \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (cosh im) (sin re)))

C

double code(double re, double im) {
	return cosh(im) * sin(re);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = cosh(im) * sin(re)
end function

Java

public static double code(double re, double im) {
	return Math.cosh(im) * Math.sin(re);
}

Python

def code(re, im):
	return math.cosh(im) * math.sin(re)

Julia

function code(re, im)
	return Float64(cosh(im) * sin(re))
end

MATLAB

function tmp = code(re, im)
	tmp = cosh(im) * sin(re);
end

Wolfram

code[re_, im_] := N[(N[Cosh[im], $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \left(e^{0 - im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \cdot \sin re \]

   7. lift-+.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right) \cdot \sin re \]

   8. +-commutative N/A

      \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \cdot \sin re \]

   9. lift-exp.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \left(\color{blue}{e^{im}} + e^{0 - im}\right)\right) \cdot \sin re \]

   10. lift-exp.f64 N/A

       \[\leadsto \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{0 - im}}\right)\right) \cdot \sin re \]

   11. lift--.f64 N/A

       \[\leadsto \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{0 - im}}\right)\right) \cdot \sin re \]

   12. sub0-neg N/A

       \[\leadsto \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{\mathsf{neg}\left(im\right)}}\right)\right) \cdot \sin re \]

   13. cosh-undef N/A

       \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(2 \cdot \cosh im\right)}\right) \cdot \sin re \]

   14. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot 2\right) \cdot \cosh im\right)} \cdot \sin re \]

   15. metadata-eval N/A

       \[\leadsto \left(\color{blue}{1} \cdot \cosh im\right) \cdot \sin re \]

   16. exp-0 N/A

       \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

   17. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

   18. exp-0 N/A

       \[\leadsto \left(\color{blue}{1} \cdot \cosh im\right) \cdot \sin re \]

   19. lower-cosh.f64 100.0

       \[\leadsto \left(1 \cdot \color{blue}{\cosh im}\right) \cdot \sin re \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]

5. Final simplification 100.0%

   \[\leadsto \cosh im \cdot \sin re \]
6. Add Preprocessing

Alternative 2: 81.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin re \cdot 0.5\\ t_1 := t\_0 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin re) 0.5)) (t_1 (* t_0 (+ (exp (- im)) (exp im)))))
   (if (<= t_1 (- INFINITY))
     (*
      (fma re (* (* re re) -0.16666666666666666) re)
      (fma (* im im) (fma im (* im 0.041666666666666664) 0.5) 1.0))
     (if (<= t_1 1.0)
       (* t_0 (fma im im 2.0))
       (*
        (fma
         im
         (fma
          (fma im (* im 0.002777777777777778) 0.08333333333333333)
          (* im (* im im))
          im)
         2.0)
        (* re 0.5))))))

C

double code(double re, double im) {
	double t_0 = sin(re) * 0.5;
	double t_1 = t_0 * (exp(-im) + exp(im));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(re, ((re * re) * -0.16666666666666666), re) * fma((im * im), fma(im, (im * 0.041666666666666664), 0.5), 1.0);
	} else if (t_1 <= 1.0) {
		tmp = t_0 * fma(im, im, 2.0);
	} else {
		tmp = fma(im, fma(fma(im, (im * 0.002777777777777778), 0.08333333333333333), (im * (im * im)), im), 2.0) * (re * 0.5);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(re) * 0.5)
	t_1 = Float64(t_0 * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(re, Float64(Float64(re * re) * -0.16666666666666666), re) * fma(Float64(im * im), fma(im, Float64(im * 0.041666666666666664), 0.5), 1.0));
	elseif (t_1 <= 1.0)
		tmp = Float64(t_0 * fma(im, im, 2.0));
	else
		tmp = Float64(fma(im, fma(fma(im, Float64(im * 0.002777777777777778), 0.08333333333333333), Float64(im * Float64(im * im)), im), 2.0) * Float64(re * 0.5));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(t$95$0 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(im * N[(N[(im * N[(im * 0.002777777777777778), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision] + 2.0), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{24}\right)} \cdot {im}^{2} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left(\sin re \cdot {im}^{2}\right)} + \sin re\right) \]

      10. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   5. Applied rewrites 73.5%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 61.3%

      \[\leadsto \mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left({im}^{2} + 2\right)} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{im \cdot im} + 2\right) \]

      3. lower-fma.f64 98.6

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Applied rewrites 98.6%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   if 1 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{im \cdot \left(im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} + 2\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right), 2\right)} \]

   5. Applied rewrites 82.1%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 69.1

         \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

   8. Applied rewrites 69.1%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{elif}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (sin re) 0.5) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma re (* (* re re) -0.16666666666666666) re)
      (fma (* im im) (fma im (* im 0.041666666666666664) 0.5) 1.0))
     (if (<= t_0 1.0)
       (sin re)
       (*
        (fma
         im
         (fma
          (fma im (* im 0.002777777777777778) 0.08333333333333333)
          (* im (* im im))
          im)
         2.0)
        (* re 0.5))))))

C

double code(double re, double im) {
	double t_0 = (sin(re) * 0.5) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(re, ((re * re) * -0.16666666666666666), re) * fma((im * im), fma(im, (im * 0.041666666666666664), 0.5), 1.0);
	} else if (t_0 <= 1.0) {
		tmp = sin(re);
	} else {
		tmp = fma(im, fma(fma(im, (im * 0.002777777777777778), 0.08333333333333333), (im * (im * im)), im), 2.0) * (re * 0.5);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(re, Float64(Float64(re * re) * -0.16666666666666666), re) * fma(Float64(im * im), fma(im, Float64(im * 0.041666666666666664), 0.5), 1.0));
	elseif (t_0 <= 1.0)
		tmp = sin(re);
	else
		tmp = Float64(fma(im, fma(fma(im, Float64(im * 0.002777777777777778), 0.08333333333333333), Float64(im * Float64(im * im)), im), 2.0) * Float64(re * 0.5));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[re], $MachinePrecision], N[(N[(im * N[(N[(im * N[(im * 0.002777777777777778), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision] + 2.0), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{24}\right)} \cdot {im}^{2} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left(\sin re \cdot {im}^{2}\right)} + \sin re\right) \]

      10. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   5. Applied rewrites 73.5%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 61.3%

      \[\leadsto \mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. lower-sin.f64 97.2

         \[\leadsto \color{blue}{\sin re} \]

   5. Applied rewrites 97.2%

      \[\leadsto \color{blue}{\sin re} \]

   if 1 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{im \cdot \left(im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} + 2\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right), 2\right)} \]

   5. Applied rewrites 82.1%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 69.1

         \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

   8. Applied rewrites 69.1%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{elif}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.2:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(re \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right), re, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) -0.2)
   (* (fma im im 2.0) (* re (fma -0.08333333333333333 (* re re) 0.5)))
   (fma (* im (* im (fma im (* im 0.041666666666666664) 0.5))) re re)))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= -0.2) {
		tmp = fma(im, im, 2.0) * (re * fma(-0.08333333333333333, (re * re), 0.5));
	} else {
		tmp = fma((im * (im * fma(im, (im * 0.041666666666666664), 0.5))), re, re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= -0.2)
		tmp = Float64(fma(im, im, 2.0) * Float64(re * fma(-0.08333333333333333, Float64(re * re), 0.5)));
	else
		tmp = fma(Float64(im * Float64(im * fma(im, Float64(im * 0.041666666666666664), 0.5))), re, re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.2], N[(N[(im * im + 2.0), $MachinePrecision] * N[(re * N[(-0.08333333333333333 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * N[(im * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * re + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.20000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left({im}^{2} + 2\right)} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{im \cdot im} + 2\right) \]

      3. lower-fma.f64 64.7

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Applied rewrites 64.7%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{12}, {re}^{2}, \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\frac{-1}{12}, \color{blue}{re \cdot re}, \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      5. lower-*.f64 33.8

         \[\leadsto \left(re \cdot \mathsf{fma}\left(-0.08333333333333333, \color{blue}{re \cdot re}, 0.5\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 33.8%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

   if -0.20000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{24}\right)} \cdot {im}^{2} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left(\sin re \cdot {im}^{2}\right)} + \sin re\right) \]

      10. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   5. Applied rewrites 91.3%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 62.4%

      \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right) \cdot re}, re\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 62.3%

       \[\leadsto \frac{1}{\frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), \left(im \cdot im\right) \cdot re, re\right)}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 65.8%

       \[\leadsto \mathsf{fma}\left(im \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right), re, re\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.2:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(re \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right), re, re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 45.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.2:\\ \;\;\;\;re \cdot \left(re \cdot \left(re \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right), re, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) -0.2)
   (* re (* re (* re -0.16666666666666666)))
   (fma (* im (* im (fma im (* im 0.041666666666666664) 0.5))) re re)))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= -0.2) {
		tmp = re * (re * (re * -0.16666666666666666));
	} else {
		tmp = fma((im * (im * fma(im, (im * 0.041666666666666664), 0.5))), re, re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= -0.2)
		tmp = Float64(re * Float64(re * Float64(re * -0.16666666666666666)));
	else
		tmp = fma(Float64(im * Float64(im * fma(im, Float64(im * 0.041666666666666664), 0.5))), re, re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.2], N[(re * N[(re * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * N[(im * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * re + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.20000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. lower-sin.f64 35.1

         \[\leadsto \color{blue}{\sin re} \]

   5. Applied rewrites 35.1%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 17.9%

      \[\leadsto \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right) \cdot -0.16666666666666666}, re\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \frac{-1}{6} \cdot {re}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 17.7%

       \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{-0.16666666666666666}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 17.7%

       \[\leadsto re \cdot \left(\left(re \cdot -0.16666666666666666\right) \cdot re\right) \]

   if -0.20000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{24}\right)} \cdot {im}^{2} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left(\sin re \cdot {im}^{2}\right)} + \sin re\right) \]

      10. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   5. Applied rewrites 91.3%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 62.4%

      \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right) \cdot re}, re\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 62.3%

       \[\leadsto \frac{1}{\frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), \left(im \cdot im\right) \cdot re, re\right)}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 65.8%

       \[\leadsto \mathsf{fma}\left(im \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right), re, re\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.2:\\ \;\;\;\;re \cdot \left(re \cdot \left(re \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right), re, re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 43.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.2:\\ \;\;\;\;re \cdot \left(re \cdot \left(re \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, re \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right), re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) -0.2)
   (* re (* re (* re -0.16666666666666666)))
   (fma (* im im) (* re (* (* im im) 0.041666666666666664)) re)))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= -0.2) {
		tmp = re * (re * (re * -0.16666666666666666));
	} else {
		tmp = fma((im * im), (re * ((im * im) * 0.041666666666666664)), re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= -0.2)
		tmp = Float64(re * Float64(re * Float64(re * -0.16666666666666666)));
	else
		tmp = fma(Float64(im * im), Float64(re * Float64(Float64(im * im) * 0.041666666666666664)), re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.2], N[(re * N[(re * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(re * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.20000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. lower-sin.f64 35.1

         \[\leadsto \color{blue}{\sin re} \]

   5. Applied rewrites 35.1%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 17.9%

      \[\leadsto \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right) \cdot -0.16666666666666666}, re\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \frac{-1}{6} \cdot {re}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 17.7%

       \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{-0.16666666666666666}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 17.7%

       \[\leadsto re \cdot \left(\left(re \cdot -0.16666666666666666\right) \cdot re\right) \]

   if -0.20000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{24}\right)} \cdot {im}^{2} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left(\sin re \cdot {im}^{2}\right)} + \sin re\right) \]

      10. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   5. Applied rewrites 91.3%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 62.4%

      \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right) \cdot re}, re\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{fma}\left(im \cdot im, \left(\frac{1}{24} \cdot {im}^{2}\right) \cdot re, re\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 62.0%

       \[\leadsto \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right) \cdot re, re\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.2:\\ \;\;\;\;re \cdot \left(re \cdot \left(re \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, re \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(re \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) 5e-15)
   (fma re (* (* re re) -0.16666666666666666) re)
   (* (* im im) (* re (* (* im im) 0.041666666666666664)))))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= 5e-15) {
		tmp = fma(re, ((re * re) * -0.16666666666666666), re);
	} else {
		tmp = (im * im) * (re * ((im * im) * 0.041666666666666664));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= 5e-15)
		tmp = fma(re, Float64(Float64(re * re) * -0.16666666666666666), re);
	else
		tmp = Float64(Float64(im * im) * Float64(re * Float64(Float64(im * im) * 0.041666666666666664)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-15], N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(re * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 4.99999999999999999e-15

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. lower-sin.f64 60.4

         \[\leadsto \color{blue}{\sin re} \]

   5. Applied rewrites 60.4%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.1%

      \[\leadsto \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right) \cdot -0.16666666666666666}, re\right) \]

   if 4.99999999999999999e-15 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{24}\right)} \cdot {im}^{2} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left(\sin re \cdot {im}^{2}\right)} + \sin re\right) \]

      10. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   5. Applied rewrites 87.2%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.6%

      \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right) \cdot re}, re\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \frac{1}{24} \cdot \left({im}^{4} \cdot \color{blue}{re}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 39.5%

       \[\leadsto \left(im \cdot im\right) \cdot \left(re \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(re \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2 \cdot 10^{-52}:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2e-52)
   (* (cosh im) (fma re (* (* re re) -0.16666666666666666) re))
   (*
    (* (sin re) 0.5)
    (fma
     im
     (fma
      (fma im (* im 0.002777777777777778) 0.08333333333333333)
      (* im (* im im))
      im)
     2.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2e-52) {
		tmp = cosh(im) * fma(re, ((re * re) * -0.16666666666666666), re);
	} else {
		tmp = (sin(re) * 0.5) * fma(im, fma(fma(im, (im * 0.002777777777777778), 0.08333333333333333), (im * (im * im)), im), 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2e-52)
		tmp = Float64(cosh(im) * fma(re, Float64(Float64(re * re) * -0.16666666666666666), re));
	else
		tmp = Float64(Float64(sin(re) * 0.5) * fma(im, fma(fma(im, Float64(im * 0.002777777777777778), 0.08333333333333333), Float64(im * Float64(im * im)), im), 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2e-52], N[(N[Cosh[im], $MachinePrecision] * N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(im * N[(N[(im * N[(im * 0.002777777777777778), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2e-52

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(e^{0 - im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \cdot \sin re \]

      7. lift-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right) \cdot \sin re \]

      8. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \cdot \sin re \]

      9. lift-exp.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\color{blue}{e^{im}} + e^{0 - im}\right)\right) \cdot \sin re \]

      10. lift-exp.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{0 - im}}\right)\right) \cdot \sin re \]

      11. lift--.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{0 - im}}\right)\right) \cdot \sin re \]

      12. sub0-neg N/A

          \[\leadsto \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{\mathsf{neg}\left(im\right)}}\right)\right) \cdot \sin re \]

      13. cosh-undef N/A

          \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(2 \cdot \cosh im\right)}\right) \cdot \sin re \]

      14. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot 2\right) \cdot \cosh im\right)} \cdot \sin re \]

      15. metadata-eval N/A

          \[\leadsto \left(\color{blue}{1} \cdot \cosh im\right) \cdot \sin re \]

      16. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

      17. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      18. exp-0 N/A

          \[\leadsto \left(\color{blue}{1} \cdot \cosh im\right) \cdot \sin re \]

      19. lower-cosh.f64 100.0

          \[\leadsto \left(1 \cdot \color{blue}{\cosh im}\right) \cdot \sin re \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]

   5. Taylor expanded in re around 0

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + re \cdot 1\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + \color{blue}{re}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\mathsf{fma}\left(re, \frac{-1}{6} \cdot {re}^{2}, re\right)} \]

      5. *-commutative N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      7. unpow2 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6}, re\right) \]

      8. lower-*.f64 72.7

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot -0.16666666666666666, re\right) \]

   7. Applied rewrites 72.7%

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)} \]

   if 2e-52 < re

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{im \cdot \left(im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} + 2\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right), 2\right)} \]

   5. Applied rewrites 93.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2 \cdot 10^{-52}:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) -0.05)
   (*
    (fma re (* (* re re) -0.16666666666666666) re)
    (fma (* im im) (fma im (* im 0.041666666666666664) 0.5) 1.0))
   (*
    (fma
     im
     (fma
      (fma im (* im 0.002777777777777778) 0.08333333333333333)
      (* im (* im im))
      im)
     2.0)
    (* re 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= -0.05) {
		tmp = fma(re, ((re * re) * -0.16666666666666666), re) * fma((im * im), fma(im, (im * 0.041666666666666664), 0.5), 1.0);
	} else {
		tmp = fma(im, fma(fma(im, (im * 0.002777777777777778), 0.08333333333333333), (im * (im * im)), im), 2.0) * (re * 0.5);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= -0.05)
		tmp = Float64(fma(re, Float64(Float64(re * re) * -0.16666666666666666), re) * fma(Float64(im * im), fma(im, Float64(im * 0.041666666666666664), 0.5), 1.0));
	else
		tmp = Float64(fma(im, fma(fma(im, Float64(im * 0.002777777777777778), 0.08333333333333333), Float64(im * Float64(im * im)), im), 2.0) * Float64(re * 0.5));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], -0.05], N[(N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(im * N[(N[(im * N[(im * 0.002777777777777778), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision] + 2.0), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.050000000000000003

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{24}\right)} \cdot {im}^{2} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left(\sin re \cdot {im}^{2}\right)} + \sin re\right) \]

      10. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   5. Applied rewrites 87.4%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 29.3%

      \[\leadsto \mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]

   if -0.050000000000000003 < (sin.f64 re)

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{im \cdot \left(im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} + 2\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right), 2\right)} \]

   5. Applied rewrites 90.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{360}, \frac{1}{12}\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 68.3

         \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]

   8. Applied rewrites 68.3%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq 10^{-54}:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, re \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right), re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) 1e-54)
   (*
    (fma re (* (* re re) -0.16666666666666666) re)
    (fma (* im im) (fma im (* im 0.041666666666666664) 0.5) 1.0))
   (fma (* im im) (* re (* (* im im) 0.041666666666666664)) re)))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= 1e-54) {
		tmp = fma(re, ((re * re) * -0.16666666666666666), re) * fma((im * im), fma(im, (im * 0.041666666666666664), 0.5), 1.0);
	} else {
		tmp = fma((im * im), (re * ((im * im) * 0.041666666666666664)), re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= 1e-54)
		tmp = Float64(fma(re, Float64(Float64(re * re) * -0.16666666666666666), re) * fma(Float64(im * im), fma(im, Float64(im * 0.041666666666666664), 0.5), 1.0));
	else
		tmp = fma(Float64(im * im), Float64(re * Float64(Float64(im * im) * 0.041666666666666664)), re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], 1e-54], N[(N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(re * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 1e-54

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{24}\right)} \cdot {im}^{2} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left(\sin re \cdot {im}^{2}\right)} + \sin re\right) \]

      10. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   5. Applied rewrites 86.3%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 65.5%

      \[\leadsto \mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]

   if 1e-54 < (sin.f64 re)

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{24}\right)} \cdot {im}^{2} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left(\sin re \cdot {im}^{2}\right)} + \sin re\right) \]

      10. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   5. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 33.7%

      \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right) \cdot re}, re\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{fma}\left(im \cdot im, \left(\frac{1}{24} \cdot {im}^{2}\right) \cdot re, re\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 33.7%

       \[\leadsto \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right) \cdot re, re\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq 10^{-54}:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, re \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{if}\;im \leq 0.35:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 4.4 \cdot 10^{+75}:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (sin re)
          (fma (* im im) (fma im (* im 0.041666666666666664) 0.5) 1.0))))
   (if (<= im 0.35)
     t_0
     (if (<= im 4.4e+75)
       (* (cosh im) (fma re (* (* re re) -0.16666666666666666) re))
       t_0))))

C

double code(double re, double im) {
	double t_0 = sin(re) * fma((im * im), fma(im, (im * 0.041666666666666664), 0.5), 1.0);
	double tmp;
	if (im <= 0.35) {
		tmp = t_0;
	} else if (im <= 4.4e+75) {
		tmp = cosh(im) * fma(re, ((re * re) * -0.16666666666666666), re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(re) * fma(Float64(im * im), fma(im, Float64(im * 0.041666666666666664), 0.5), 1.0))
	tmp = 0.0
	if (im <= 0.35)
		tmp = t_0;
	elseif (im <= 4.4e+75)
		tmp = Float64(cosh(im) * fma(re, Float64(Float64(re * re) * -0.16666666666666666), re));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.35], t$95$0, If[LessEqual[im, 4.4e+75], N[(N[Cosh[im], $MachinePrecision] * N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if im < 0.34999999999999998 or 4.40000000000000024e75 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{24}\right)} \cdot {im}^{2} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left(\sin re \cdot {im}^{2}\right)} + \sin re\right) \]

      10. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   5. Applied rewrites 94.5%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   if 0.34999999999999998 < im < 4.40000000000000024e75

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(e^{0 - im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \cdot \sin re \]

      7. lift-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right) \cdot \sin re \]

      8. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \cdot \sin re \]

      9. lift-exp.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\color{blue}{e^{im}} + e^{0 - im}\right)\right) \cdot \sin re \]

      10. lift-exp.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{0 - im}}\right)\right) \cdot \sin re \]

      11. lift--.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{0 - im}}\right)\right) \cdot \sin re \]

      12. sub0-neg N/A

          \[\leadsto \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{\mathsf{neg}\left(im\right)}}\right)\right) \cdot \sin re \]

      13. cosh-undef N/A

          \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(2 \cdot \cosh im\right)}\right) \cdot \sin re \]

      14. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot 2\right) \cdot \cosh im\right)} \cdot \sin re \]

      15. metadata-eval N/A

          \[\leadsto \left(\color{blue}{1} \cdot \cosh im\right) \cdot \sin re \]

      16. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

      17. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      18. exp-0 N/A

          \[\leadsto \left(\color{blue}{1} \cdot \cosh im\right) \cdot \sin re \]

      19. lower-cosh.f64 100.0

          \[\leadsto \left(1 \cdot \color{blue}{\cosh im}\right) \cdot \sin re \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]

   5. Taylor expanded in re around 0

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + re \cdot 1\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + \color{blue}{re}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\mathsf{fma}\left(re, \frac{-1}{6} \cdot {re}^{2}, re\right)} \]

      5. *-commutative N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      7. unpow2 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6}, re\right) \]

      8. lower-*.f64 83.3

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot -0.16666666666666666, re\right) \]

   7. Applied rewrites 83.3%

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.35:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{elif}\;im \leq 4.4 \cdot 10^{+75}:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 83.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sin re \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{if}\;im \leq 0.0145:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 7 \cdot 10^{+88}:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{elif}\;im \leq 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(im, re \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (sin re) 0.5) (fma im im 2.0))))
   (if (<= im 0.0145)
     t_0
     (if (<= im 7e+88)
       (* (cosh im) (fma re (* (* re re) -0.16666666666666666) re))
       (if (<= im 1e+154)
         (fma im (* re (* im (fma im (* im 0.041666666666666664) 0.5))) re)
         t_0)))))

C

double code(double re, double im) {
	double t_0 = (sin(re) * 0.5) * fma(im, im, 2.0);
	double tmp;
	if (im <= 0.0145) {
		tmp = t_0;
	} else if (im <= 7e+88) {
		tmp = cosh(im) * fma(re, ((re * re) * -0.16666666666666666), re);
	} else if (im <= 1e+154) {
		tmp = fma(im, (re * (im * fma(im, (im * 0.041666666666666664), 0.5))), re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(sin(re) * 0.5) * fma(im, im, 2.0))
	tmp = 0.0
	if (im <= 0.0145)
		tmp = t_0;
	elseif (im <= 7e+88)
		tmp = Float64(cosh(im) * fma(re, Float64(Float64(re * re) * -0.16666666666666666), re));
	elseif (im <= 1e+154)
		tmp = fma(im, Float64(re * Float64(im * fma(im, Float64(im * 0.041666666666666664), 0.5))), re);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.0145], t$95$0, If[LessEqual[im, 7e+88], N[(N[Cosh[im], $MachinePrecision] * N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1e+154], N[(im * N[(re * N[(im * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.0145000000000000007 or 1.00000000000000004e154 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left({im}^{2} + 2\right)} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{im \cdot im} + 2\right) \]

      3. lower-fma.f64 85.5

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Applied rewrites 85.5%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   if 0.0145000000000000007 < im < 6.9999999999999995e88

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(e^{0 - im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \cdot \sin re \]

      7. lift-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right) \cdot \sin re \]

      8. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \cdot \sin re \]

      9. lift-exp.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\color{blue}{e^{im}} + e^{0 - im}\right)\right) \cdot \sin re \]

      10. lift-exp.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{0 - im}}\right)\right) \cdot \sin re \]

      11. lift--.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{0 - im}}\right)\right) \cdot \sin re \]

      12. sub0-neg N/A

          \[\leadsto \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{\mathsf{neg}\left(im\right)}}\right)\right) \cdot \sin re \]

      13. cosh-undef N/A

          \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(2 \cdot \cosh im\right)}\right) \cdot \sin re \]

      14. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot 2\right) \cdot \cosh im\right)} \cdot \sin re \]

      15. metadata-eval N/A

          \[\leadsto \left(\color{blue}{1} \cdot \cosh im\right) \cdot \sin re \]

      16. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

      17. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      18. exp-0 N/A

          \[\leadsto \left(\color{blue}{1} \cdot \cosh im\right) \cdot \sin re \]

      19. lower-cosh.f64 100.0

          \[\leadsto \left(1 \cdot \color{blue}{\cosh im}\right) \cdot \sin re \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]

   5. Taylor expanded in re around 0

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + re \cdot 1\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + \color{blue}{re}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\mathsf{fma}\left(re, \frac{-1}{6} \cdot {re}^{2}, re\right)} \]

      5. *-commutative N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      7. unpow2 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6}, re\right) \]

      8. lower-*.f64 85.0

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot -0.16666666666666666, re\right) \]

   7. Applied rewrites 85.0%

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)} \]

   if 6.9999999999999995e88 < im < 1.00000000000000004e154

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{24}\right)} \cdot {im}^{2} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left(\sin re \cdot {im}^{2}\right)} + \sin re\right) \]

      10. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 85.6%

      \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right) \cdot re}, re\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 85.6%

       \[\leadsto \frac{1}{\frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), \left(im \cdot im\right) \cdot re, re\right)}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 100.0%

       \[\leadsto \mathsf{fma}\left(im, \left(im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right) \cdot \color{blue}{re}, re\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0145:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 7 \cdot 10^{+88}:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{elif}\;im \leq 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(im, re \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 52.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq -0.05:\\ \;\;\;\;re \cdot \left(re \cdot \left(re \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, re \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right), re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) -0.05)
   (* re (* re (* re -0.16666666666666666)))
   (fma im (* re (* im (fma im (* im 0.041666666666666664) 0.5))) re)))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= -0.05) {
		tmp = re * (re * (re * -0.16666666666666666));
	} else {
		tmp = fma(im, (re * (im * fma(im, (im * 0.041666666666666664), 0.5))), re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= -0.05)
		tmp = Float64(re * Float64(re * Float64(re * -0.16666666666666666)));
	else
		tmp = fma(im, Float64(re * Float64(im * fma(im, Float64(im * 0.041666666666666664), 0.5))), re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], -0.05], N[(re * N[(re * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(im * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.050000000000000003

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. lower-sin.f64 48.1

         \[\leadsto \color{blue}{\sin re} \]

   5. Applied rewrites 48.1%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 23.8%

      \[\leadsto \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right) \cdot -0.16666666666666666}, re\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \frac{-1}{6} \cdot {re}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 23.8%

       \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{-0.16666666666666666}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 23.8%

       \[\leadsto re \cdot \left(\left(re \cdot -0.16666666666666666\right) \cdot re\right) \]

   if -0.050000000000000003 < (sin.f64 re)

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{24}\right)} \cdot {im}^{2} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left(\sin re \cdot {im}^{2}\right)} + \sin re\right) \]

      10. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   5. Applied rewrites 88.4%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 62.7%

      \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right) \cdot re}, re\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 62.6%

       \[\leadsto \frac{1}{\frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), \left(im \cdot im\right) \cdot re, re\right)}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 64.7%

       \[\leadsto \mathsf{fma}\left(im, \left(im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right) \cdot \color{blue}{re}, re\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.05:\\ \;\;\;\;re \cdot \left(re \cdot \left(re \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, re \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 47.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq -0.05:\\ \;\;\;\;re \cdot \left(re \cdot \left(re \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, re \cdot 0.5, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) -0.05)
   (* re (* re (* re -0.16666666666666666)))
   (fma (* im im) (* re 0.5) re)))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= -0.05) {
		tmp = re * (re * (re * -0.16666666666666666));
	} else {
		tmp = fma((im * im), (re * 0.5), re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= -0.05)
		tmp = Float64(re * Float64(re * Float64(re * -0.16666666666666666)));
	else
		tmp = fma(Float64(im * im), Float64(re * 0.5), re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], -0.05], N[(re * N[(re * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(re * 0.5), $MachinePrecision] + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.050000000000000003

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. lower-sin.f64 48.1

         \[\leadsto \color{blue}{\sin re} \]

   5. Applied rewrites 48.1%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 23.8%

      \[\leadsto \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right) \cdot -0.16666666666666666}, re\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \frac{-1}{6} \cdot {re}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 23.8%

       \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{-0.16666666666666666}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 23.8%

       \[\leadsto re \cdot \left(\left(re \cdot -0.16666666666666666\right) \cdot re\right) \]

   if -0.050000000000000003 < (sin.f64 re)

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{24}\right)} \cdot {im}^{2} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left(\sin re \cdot {im}^{2}\right)} + \sin re\right) \]

      10. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   5. Applied rewrites 88.4%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 62.7%

      \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right) \cdot re}, re\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(im \cdot im, \frac{1}{2} \cdot re, re\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 57.0%

       \[\leadsto \mathsf{fma}\left(im \cdot im, 0.5 \cdot re, re\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.05:\\ \;\;\;\;re \cdot \left(re \cdot \left(re \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, re \cdot 0.5, re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 34.0% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (fma re (* (* re re) -0.16666666666666666) re))

C

double code(double re, double im) {
	return fma(re, ((re * re) * -0.16666666666666666), re);
}

Julia

function code(re, im)
	return fma(re, Float64(Float64(re * re) * -0.16666666666666666), re)
end

Wolfram

code[re_, im_] := N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\sin re} \]
4. Step-by-step derivation

   1. lower-sin.f64 51.1

      \[\leadsto \color{blue}{\sin re} \]

5. Applied rewrites 51.1%

   \[\leadsto \color{blue}{\sin re} \]

6. Taylor expanded in re around 0

   \[\leadsto re \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 34.4%

   \[\leadsto \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right) \cdot -0.16666666666666666}, re\right) \]
9. Add Preprocessing

Alternative 16: 10.6% accurate, 19.8× speedup

Math

\[\begin{array}{l} \\ re \cdot \left(re \cdot \left(re \cdot -0.16666666666666666\right)\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* re (* re (* re -0.16666666666666666))))

C

double code(double re, double im) {
	return re * (re * (re * -0.16666666666666666));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re * (re * (re * (-0.16666666666666666d0)))
end function

Java

public static double code(double re, double im) {
	return re * (re * (re * -0.16666666666666666));
}

Python

def code(re, im):
	return re * (re * (re * -0.16666666666666666))

Julia

function code(re, im)
	return Float64(re * Float64(re * Float64(re * -0.16666666666666666)))
end

MATLAB

function tmp = code(re, im)
	tmp = re * (re * (re * -0.16666666666666666));
end

Wolfram

code[re_, im_] := N[(re * N[(re * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\sin re} \]
4. Step-by-step derivation

   1. lower-sin.f64 51.1

      \[\leadsto \color{blue}{\sin re} \]

5. Applied rewrites 51.1%

   \[\leadsto \color{blue}{\sin re} \]

6. Taylor expanded in re around 0

   \[\leadsto re \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 34.4%

   \[\leadsto \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right) \cdot -0.16666666666666666}, re\right) \]

9. Taylor expanded in re around inf

   \[\leadsto \frac{-1}{6} \cdot {re}^{\color{blue}{3}} \]
10. Step-by-step derivation

11. Applied rewrites 11.0%

    \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{-0.16666666666666666}\right) \]
12. Step-by-step derivation

13. Applied rewrites 11.0%

    \[\leadsto re \cdot \left(\left(re \cdot -0.16666666666666666\right) \cdot re\right) \]

14. Final simplification 11.0%

    \[\leadsto re \cdot \left(re \cdot \left(re \cdot -0.16666666666666666\right)\right) \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024234 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))